Creating Ratios

Ratios can look complicated, but they are just fractions that show the relationship between two numbers that increase and decrease together.  You can write ratios with colons (6:5) or as fractions 6/5. 

The only tricky thing to remember about ratios is that you have to be very careful about the order of your numbers.  The way to say or write a ratio makes a huge impact on what it means.  So, if you have a classroom with 1 teacher and 30 kids, it's important that you write the student:teacher ratio as 30:1.  If you write the student:teacher ratio as 1:30, it means you have thirty teachers and only one student! 

There are some basic facts to know about ratios:

• Ratios show how numbers are related.  When one number in a ratio goes up, the other number goes up too.  When one number goes down, the other goes down too!
• Ratios can be written as fractions and should be reduced or simplified like fractions.  So, if the student:teacher ratio in a school is 600:30, you can write it like $\dfrac{600}{30}$ and reduce it to $\dfrac{20}{1} \text{ or } 20:1$.
• Order matters.  The first number in a ratio (or the top number, if you are writing your ratio like a fraction) is the first item stated in the ratio. 
• Sometimes you will have to do a calculation to figure out one of the numbers in a ratio (for instance, finding a total).

Example:

Let's say that a baseball team wins 15 games in a season.  They lose ten games.  They do not tie any games.

What is the ratio of the team's wins to losses?

It often helps to start out by writing a ratio in words.  That helps you know which number to put in which place in the fraction.  Then substitute the words for numbers.  Then reduce.

$\dfrac{\text{wins}}{\text{losses}}=\dfrac{15}{10}=\dfrac{3}{2}$

What is the ratio of the team's loses to wins?

Remember, order matters.  Switches like this show why it often helps to start out by writing a ratio in words.  Substitute the words for numbers.  Then reduce.

$\dfrac{\text{losses}}{\text{wins}}=\dfrac{10}{15}=\dfrac{2}{3}$

What is the ratio of the team's wins to total games?

Writing out the words can help you figure out what you know and what you don't know.  In this case, you were not given total games.  You have to add wins and losses together.  Then proceed as usual.

$\dfrac{\text{wins}}{\text{total games}}=\dfrac{15}{15+10}=\dfrac{15}{25}=\dfrac{3}{5}$

Note, here, the ratio also helps you make a little percentage.  They won $\dfrac{3}{5}$ of their games or 60% of their games. 

Overall, ratios are basic (and are usually just the first step in a more complicated question.  Just make sure to pay attention and be meticulous about matching up items and numbers (you may have to figure out numbers that are not given in the problem!).